In this paper, we study the relation between the uniform Roe algebra and the uniform quasi-local algebra associated to a metric space of bounded geometry. In the process, we introduce a weakening of the notion of expanders, called asymptotic expanders. We show that being a sequence of asymptotic expanders is a coarse property under certain connectedness condition, and it implies non-uniformly local amenability. Moreover, we also analyse some $C*$-algebraic properties of uniform quasi-local algebras. In particular, we show that a uniform quasi-local algebra is nuclear if and only if the underlying metric space has Property A.

中文翻译：

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准局部代数和渐近扩展器

在本文中，我们研究了均匀 Roe 代数和与有界几何的度量空间相关联的均匀拟局部代数之间的关系。在这个过程中，我们引入了弱化扩展器的概念，称为渐近扩展器。我们表明，在一定连通性条件下，渐近扩展器序列是一个粗略的性质，它意味着非均匀的局部适应性。此外，我们还分析了一致准局部代数的一些$C*$-代数性质。特别是，当且仅当基础度量空间具有属性 A 时，我们证明了一致准局部代数是核的。